Primary resonance of traveling viscoelastic beam under internal resonance. (English) Zbl 1358.74017
Summary: Under the 3:1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The undetermined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.
MSC:
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74H10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics |
Keywords:
traveling beam; nonlinear vibration; viscoelasticity; primary resonance; internal resonanceReferences:
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